We study the complexity of Banach space valued integration problems in the randomized setting. We explain the interplay between the geometry of the underlying Banach space and the rate of the nn-th minimal errors. The relevant notion of Banach space geometry is the Rademacher type of the space. We also explain the motivation which comes from applications to the complexity of definite and indefinite integration and to the solution of ODEs. This is joint work with Stefan Heinrich.